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Markkula, G orcid.org/0000-0003-0244-1582, Benderius, O and Wahde, M (2014) Comparing and validating models of driver steering behaviour in collision avoidance and vehicle stabilisation. Vehicle System Dynamics, 52 (12). 12. pp. 1658-1680. ISSN 0042-3114

https://doi.org/10.1080/00423114.2014.954589

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August 10, 2014 21:47 Vehicle System Dynamics MarkkulaEtAlVSD

RESEARCH ARTICLE

Comparing and validating models of driver steering behaviour in

collision avoidance and vehicle stabilization

G. Markkulaa,b,∗, O. Benderiusb, and M. Wahdeb

aTransport Analysis, Volvo Group Trucks Technology, Advanced Technology and

Research, Göteborg, Sweden; bDepartment of Applied Mechanics, Chalmers University of

Technology, Göteborg, Sweden

(Received Month Day Year)

A number of driver models were fitted to a large data set of human truck driving, from asimulated near-crash, low-friction scenario, yielding two main insights: Steering to avoid acollision was best described as an open-loop manoeuvre of predetermined duration, but withsituation-adapted amplitude, and subsequent vehicle stabilization could to a large extent beaccounted for by a simple yaw rate nulling control law. These two phenomena, which couldbe hypothesized to generalize to passenger car driving, were found to determine the abilityof four driver models adopted from literature to fit the human data. Based on the obtainedresults, it is argued that the concept of internal vehicle models may be less valuable whenmodelling driver behaviour in non-routine situations such as near-crashes, where behaviourmay be better described as direct responses to salient perceptual cues. Some methodologicalissues in comparing and validating driver models are also discussed.

Keywords: Driver models, steering, collision avoidance, low friction, driving experience,electronic stability control

Manuscript word count: 7501 words including appendix

1. Introduction

It is well established that driver behaviour plays a prominent role in the causationof traffic accidents [1, 2], and considerable research effort has been spent on un-derstanding and describing driver behaviour in near-crash situations. This is notan easy object of study, but as a result of accident reconstructions, large-scalenaturalistic data collection projects, and experiments on test tracks and in drivingsimulators, there is a growing body of knowledge on the various reasons why driversend up in critical situations, such as inattention [3] or incorrect expectations [4, 5],and on how drivers typically control the vehicle if and when they try to avoid animminent crash [6–9].An important application of such knowledge is the construction of quantitative

models of driver control behaviour in near-crash situations. When put to use incomputer simulations, such models permit cost-efficient safety performance opti-mization of, for example, infrastructure designs [10], vehicle designs [11], or activesupport systems that provide warnings or control interventions [12–14].

∗Corresponding author. E-mail: [email protected]

ISSN: 0042-3114 print/ISSN 1744-5159 onlinec⃝ 201x Taylor & FrancisDOI: 10.1080/0042311YYxxxxxxxxhttp://www.informaworld.com

(Final draft post-refereeing, pre-publication)


2 G. Markkula et al.

There is a wealth of existing driver behaviour models that could be useful in suchsimulation-based research efforts, reviewed in [15–19]. However, a recent review, fo-cusing specifically on models that have been applied in simulation of near-collisionsituations [20], noted two clear limitations in the literature: (a) With just a fewexceptions [21–23], new models have been proposed without comparing their be-haviour to that of existing, alternative models, making it difficult to know whichmodels to prefer for a given application. (b) Validation of model behaviour againsthuman behaviour data from real or reasonably realistic near-crash situations hasbeen virtually non-existent. Many models of steering control were found to havebeen validated against human behaviour in predefined test-track manoeuvres, forexample a double lane change [24]. However, such tests seem rather unlike real-life,unexpected near-crash situations, and could potentially elicit qualitatively differentbehaviours from drivers [25–27].This paper addresses both of the two above-mentioned limitations, in the specific

context of collision avoidance and subsequent vehicle stabilization, on a low frictionroad surface. One use for models validated in this type of context is simulation-based evaluation of vehicle stability support systems such as electronic stabilitycontrol (ESC) [28–31]. In [32], it was shown that one existing driver model couldreproduce the stabilization steering behaviour observed after unexpected and ex-pected near-collisions in a driving simulator study, previously described in [33].Here, the analysis of this dataset, fitting models separately to each human driver,will be extended to include also the collision avoidance phase of the studied sce-nario, and to include a comparison of a number of existing and novel models ofsteering.The next section will describe the data collection simulator study and the driver

models, as well as the method for fitting the models to the human steering data.Then, model-fitting results will be provided, including some analysis of the obtainedmodel parameters. The subsequent discussion will highlight differences between themodels and their respective strengths and weaknesses in the studied scenario, aswell as some challenges involved in model comparison and validation.

2. Method

2.1. Data collection

The human driving data used here were collected in the moving-base driving sim-ulator VTI Simulator II in Linköping, Sweden. Full details on the simulator andthe experimental procedures adopted in this study can be found in [33]. In sum-mary, 48 drivers, driving a three-axle rigid simulated truck (6.2 m from first to lastaxle) at 80 km/h, experienced an unexpected lead vehicle deceleration scenario ona low-friction (µ = 0.25) road surface. Half of the subject drivers subsequently alsoexperienced the same scenario an additional twelve times each, in a novel paradigmfor repeated collision avoidance, and it is this 24-driver repeated-scenario data setwhich is used here. Half of the 24 drivers were novices, who had just obtained,or were just about to obtain, their heavy truck driving license, and half were ex-perienced drivers, with at least six years of professional experience in commercialoperations. In half of all measurements, the simulated truck had an active ESC sys-tem, a software-in-the-loop implementation of the actual Volvo Trucks on-marketESC, and in the other half of measurements drivers were aided only by the anti-lock braking system (ABS). The critical scenario, requiring a steering manoeuvrefor successful collision avoidance, is illustrated in Fig. 1, together with an overviewof the observed vehicle trajectories.


Vehicle System Dynamics 3

(a)

−100 −50 0 50 100 150 200

−10

−8

−6

−4

−2

Longitudinal position (m)

Late

ral p

ositi

on (

m)

−100 −50 0 50 100 150 200

−10

−8

−6

−4

−2


Late

ral p

ositi

on (

m)

(b)

Figure 1. The studied critical scenario. (a) A screenshot from the experiment video log, showing the two-lane winter road on which the maneuvering took place, the braking lead vehicle, and a subject engagedin steering collision avoidance. (b) The totality of observed vehicle trajectories, in the unexpected (toppanel) and repeated (bottom panel) scenarios. Horizontal gray lines show lane boundaries, and each blackline shows the movement of the front center of the truck in one recorded event. Longitudinal position zerocorresponds to the point where the front of the truck reached the rear of the lead vehicle. The unexpectedscenario data are not used in the model-fitting analyses presented in this paper, but are shown here toillustrate that the two scenarios generated roughly similar human behaviour; see further [32] and [34].

It has been demonstrated elsewhere that the unexpected and repeated scenariosgenerated similar initial steering avoidance situations [33, 34], and elicited similardriving steering behaviour, both during collision avoidance [34] and stabilization[32].

2.2. Tested models

The set of driver models to test was defined so as to include both some well-known path-following models of steering, often available in off-the-shelf softwarefor e.g. simulating predefined manoeuvres (the MacAdam and Sharp et al. models),as well as models of routine lane-keeping which may be less familiar but whichtake different, and in our view promising, modelling approaches (the Salvucci &Gray and Gordon & Magnuski models). Furthermore, based on the results fromparameter-fitting these existing models to the behaviour of the human drivers, twovery simple additional models of steering were developed, one targeting collisionavoidance only, and the other targeting only vehicle stabilization. Below, all testedmodels will be briefly described, along with specific implementation details whenneeded. For ease of reading, consistent notation is used for quantities that areshared across models, in some cases departing from the symbols used by the originalauthors.

2.2.1. The MacAdam model

At a given time t, the model proposed by MacAdam [35], illustrated in Fig. 3(a),applies the steering wheel angle δ(t) that minimizes the predicted lateral deviationfrom a desired path, by minimizing the following functional:

J(t) =

∫ t−TR+TP

t−TR

(f(η)− y(η))2 dη (1)



XS

1

X2

X3

X4

Y

-50 0 50 100 150 200

-10

-8

-6

-4

-2


Late

ral p

ositi

on (

m)

Figure 2. The desired path used for the MacAdam [35] and Sharp et al. [36] models, with ∆X1, X2,X3, X4, and Y as driver-specific model parameters. The two lane changes are cubic splines with lateralspeed zero at beginning and end. Before and after the manoeuvre, desired lateral position is set to themiddle of the right driving lane. Longitudinal position zero corresponds to the point where the front ofthe truck reached the rear of the lead vehicle, and XS is the longitudinal position at which the humandriver’s collision avoidance steering reached half of its maximum value, in a specific recorded instance ofthe critical scenario.

In Eq. (1), y(t) and f(t) are the predicted and desired lateral positions, and TRand TP are model parameters corresponding to a reaction time and a preview time,respectively. Here, the desired path f(t) of the vehicle was defined as shown inFig. 2, using model parameters ∆X1, X2, X3, X4, and Y . In order to allow forintra-driver variability in the exact point of collision avoidance initiation, the lanechange to the left was set to start at a longitudinal position XS − ∆X1, whereXS was the longitudinal position at which the steering wheel reached half of itsmaximum leftward deflection (i.e.XS had a unique value for every recorded instanceof the critical scenario).The MacAdam model’s prediction y(t) of lateral position relies on a linear in-

ternal vehicle model. Here, the same classical type of one-track model as used byMacAdam [35] was adopted, but with three axles instead of two:

ẋ = Fx+ gδ =

[

−Cαf+Cαm+Cαrmvx

−aCαf+bCαm+cCαrmvx

− vx−aCαf+bCαm+cCαr

Izvx−a

2Cαf+b2Cαm+c2CαrIzvx

]

x+

[

GCαfm

GafCαfIz

]

δ (2)

where x = [vy ψ̇]T , vx and vy are longitudinal and lateral speeds in the vehicle’s

reference frame, and ψ is the yaw angle of the vehicle. The three Cα � parametersand a, b, c are tire cornering stiffnesses and longitudinal distances to the vehicle’smass centre, for the front, middle, and rear axles, respectively. The parameters mand Iz are vehicle mass and moment of inertia, and G is the steering gear ratio.Since the linear vehicle model cannot account well for skidding, it was parameter-

fitted only to recordings with maximum body slip angle β < 1◦ (3 % of the totaldata set). This can be understood as assuming that drivers had acquired an under-standing of vehicle dynamics from normal, high-friction driving, and applied thisunderstanding also during yaw instability1. Only the three cornering stiffnesseswere fitted to the data; the other parameters were taken from the non-linear modelused in the simulator study. Fig. 4 illustrates the resulting model performance atvarious magnitudes of yaw instability.

2.2.2. The Sharp et al. model

The model proposed by Sharp et al. [36] also makes use of the desired path con-struct, but, as illustrated schematically in Fig. 3(b), instead calculates its steeringwheel input as a weighted sum of current and previewed path deviations ei along aforward optical lever, extending a preview time TP ahead, and the current deviation

1Various approaches were explored for fitting the linear model also to recordings with more severe yawinstability, but were not found to improve the fit of the resulting driver model.



Desired path

Computed

optimal path

(a) MacAdam [35]

Desired path

Optical lever

(b) Sharp et al. [36]

Target

lat. pos.

Near point Far point

X3

(c) Salvucci & Gray [37]

Current path

Required path

Boundary points

X4

(d) Gordon & Magnuski [38]

Figure 3. Schematic illustrations of the models adopted from literature.

0 10 20−100

0

100

δ (°

)

β ≤ 3°

0 10 20−5

0

5

Lat.

acc.

(m/s

2 )

0 10 20−50

0

50

Yaw

acc

.(°

/s2 )

Time(s)

0 10 20−500

0

500β ≤ 11°

0 10 20−10

0

10

0 10 20−200

0

200

Time(s)

0 10 20−500

0

500β ≤ 16°

0 10 20−10

0

10

0 10 20−200

0

200

Time(s)

Simulator studyLinear veh. model

Figure 4. The top row of panels show observed steering wheel movements in three recordings of therepeated scenario, with maximum attained body slip angles β increasing from left to right. The bottomtwo rows of panels show, for the same recordings, the observed vehicle dynamics from the full non-linearvehicle dynamics model used in the data collection experiment, compared with the vehicle dynamicspredicted by the linear model used with the MacAdam driver model in this paper.

eψ between vehicle and path heading:

δ = Kψeψ +K1e1 +Kp

n∑

i=2

Kiei (3)

Here, Kψ, K1, and Kp were treated as free model parameters, whereas the numbern of preview points, their spacing along the optical lever, and the exponentiallydecreasing profile for the preview gains Ki (with 2 ≤ i ≤ n) were adopted from[36]. Additionally, to allow for a fair comparison with the other models, a reactiontime delay parameter TR was added to Eq. (3). The saturation functions includedby Sharp et al. with the purpose of “preventing the steer angle from exceeding areasonable range”[36, p. 312], were not included.



2.2.3. The Salvucci and Gray model

The model by Salvucci & Gray [37] is mathematically rather similar to the Sharpet al. model. However, instead of being derived from linear optimal control theory,it builds on experiments and modelling in psychology, motivating: (a) the use ofthe rate of change δ̇ of steering as an input variable rather than δ [39], and (b)the separation of controlled quantities (the input to the driver) into one near pointand one far point [40] on a target lateral position, as illustrated in Fig. 3(c). It isassumed that the driver aims to keep the the sight angles θn and θf to these pointsstationary, while at the same time attempting to reduce the near point angle tozero:

δ̇ = knPθ̇n + kf θ̇f + knIθn (4)

In addition to the gain parameters in Eq. (4), free parameters were also included forthe longitudinal distances Dn and Df to the near and far points, respectively

1, aswell as a reaction time TR. Analogously to the desired path of the models describedabove, the target lateral position was initially set to the middle of the right lane,then set to a position Y when the truck reached longitudinal position XS +∆XI,and then back to the middle of the right lane at longitudinal position X3. Totest the sensitivity of the model to the preview distance parameters, an additionalversion of the model was tested, where these parameters were fixed, for all drivers,at the median values Dn = 16 m and Df = 123 m, observed in the optimizationswhere these parameters were left free.

2.2.4. The Gordon and Magnuski model

Since the models described above all aim at reducing the deviation from a desiredpath or lateral position to zero, they could be referred to as optimizing models. Incontrast, the model by Gordon & Magnuski [38], illustrated in Fig. 3(d), operatesin what can be called a satisficing [41] manner: It assumes that the driver is contentwith staying inside a delimited region, modelled using boundary points. Specifically,the model compares the current yaw rate to the yaw rates needed to steer clear ofeach boundary point, identifies the point with the greatest mismatch, and appliesa rate of steering aimed at reaching, within a time τs, the required yaw rate ψ̇reqfor this point, assuming a simple vehicle model with wheel base L:

δ̇ = −L

Gτsvx(ψ̇ − ψ̇req) (5)

Before computing ψ̇req, the model also applies a vehicle state prediction to coun-teract its own reaction time delay TR.The original publication [38] considered only lane keeping, but Chang [42] ap-

plied the same model to avoidance of static obstacles, with a safety margin ρC. Inthe present work, seemingly the first time the model is applied to obstacles andlane boundaries simultaneously, conflicts with lead vehicle boundary points weregiven priority over lane boundary conflicts, and a separate safety margin ρL forlane boundaries was added, with allowed negative values in the optimization, toaccount for the apparent acceptance of moderate lane excursions in some of the

1Following [40], Salvucci & Gray [37] specified preview in terms of angles down from the horizon, whichin practice amounts to the same as using a preview distance. Here, it was also attempted to make thepreview speed-dependent, as in the MacAdam and Sharp et al. models, e.g. Dn = Tnvx, but if anythingthis reduced the model’s ability of fitting the human data.



0 50 100 150 200 250 3000

100

200

300

400

500

600

700

Max. steering wheel angle (º)

Max

. ste

erin

g w

heel

ang

le r

ate

(º/s

)

Unexpected (r = 0.90, k = 2.41, p < 0.0001)Repeated (r = 0.88, k = 2.21)

(a)

−0.42 0 0.42

0

1

Time (s)

St.

wh.

ang

le (

º)

−0.42 0 0.42

0

2.21

St.

wh.

ang

le r

ate

(º/s

)

Time (s)

(b)

Figure 5. Open loop avoidance steering. (a) The observed correlation between maximum leftward steeringwheel angle before reaching the lead vehicle, and maximum leftward steering wheel angle rate during thesame period. The Pearson correlation coefficients r are provided, as well as the slope k of least-squares fitlines with zero intercept. The unexpected scenario data are not used in the model-fitting analyses presentedin this paper, but are shown here to illustrate that the two scenarios generated roughly similar humanbehaviour; see further [32] and [34]. (b) The steering wheel input generated by the open loop avoidancemodel in this paper, here parameterized to illustrate the width of such a steering wheel pulse (0.84 s) thatwould yield the k observed for the repeated scenario in panel (a).

human drivers. Furthermore, extending the model to handle the non-static lead ve-hicle, lead vehicle state prediction was included, as also illustrated schematically inFig. 3(d). In three different versions of the model, this prediction was done assum-ing a constant lead vehicle acceleration (2nd order), speed (1st order), and position(0th order), respectively. To implement the scenario studied here, the lead vehicleboundary points were included only from longitudinal position XS+∆XI, and thelane change back to the right lane was achieved by placing the left-side boundarypoints beyond longitudinal position X4 to between the two driving lanes; again,see Fig. 3(d).

2.2.5. Open loop avoidance models

An additional model of collision avoidance steering was tested, motivated by alinear correlation previously reported by Breuer [25], and replicated here: As shownin Fig. 5(a), higher-amplitude avoidance manoeuvres were carried out with fastersteering movements. This finding suggests (a) that avoidance manoeuvre durationwas roughly constant between scenario recordings, and (b) that each manoeuvre’samplitude was determined before its initiation.Therefore, in contrast to the closed-loop models described above, which calculate

a new control input at each time step in a simulation, an open-loop model wasposited, applying a pulse of steering wheel rotation represented as a Gaussian cutoff at ±2 standard deviations; see Fig. 5(b). The pulse duration TD = 2TH wasincluded as a free parameter, and pulse amplitude was determined as a functionof the collision situation a reaction time TR before manoeuvre initiation. To allowfor the above-mentioned intra-driver variation in collision avoidance timing, thepeak of the pulse was placed at time TS + TA, where TS was the time at which thetruck’s longitudinal position was XS (see above), and TA was another free modelparameter.Five different versions of the model were tested, all using a model parameter K

to determine the steering pulse amplitude as (a) a constant, situation-independent



amplitude K, (b) K times the optical expansion, or looming, of the lead vehicleon the driver’s retina [43], or (c-e) K times the steering required to avoid the leadvehicle with a safety margin ρC , given the steady state yaw rate response of thelinear vehicle model described in 2.2.1, and a lead vehicle state prediction of orderszero through two (cf. Sec. 2.2.4).While this model is neutral on whether the driver is controlling steering wheel

angle δ or its rate of change δ̇, the choice of target signal has an impact on modelparameter-fitting (further discussed in Sect. 4.3). Therefore, this model was fittedseparately both as controlling δ and δ̇.The model’s sensitivity to the TR parameter was also tested, by parameter-fitting

an additional version of the δ-controlling, 2nd order yaw rate requirement model,with TR fixed at 0.2 s for all drivers.

2.2.6. Yaw angle/rate nulling stabilization models

As has been reported elsewhere [32], the Salvucci & Gray model is reasonablysuccessful at fitting the stabilization steering data studied here. Additional explo-ration indicated that much of the variance explained by the model was accountedfor by its far point control (the second term in Eq. 4), shown in [32] to approximatea yaw rate nulling steering behaviour: δ̇ = −Kψ̇, where K is a model parameter.Here, such a model was tested directly, as well as a time-integrated yaw anglenulling version δ = −Kψ, both with a reaction time delay TR.

2.3. Division into avoidance and stabilization steering phases

Preliminary experimentation indicated that the steering models were differentiallysuccessful at fitting steering during collision avoidance and vehicle stabilization.Therefore, the data set was split accordingly, and model parameter-fitting wascarried out separately on the two sets.The collision avoidance phase of a recorded scenario was defined to begin when

the lead vehicle started decelerating, and to end when the driver began applyingconsiderable rightward steering wheel rotation, interpretable as a transition fromleftward collision avoidance, to lane alignment and vehicle stabilization. This onsetof rightward steering wheel rotation was generally clearly visible in the data, andwas found to be suitably defined as the last point of leftward steering (δ > 0)where δ̇ > −50◦/s. The example recordings in Fig. 6 (further explained in Sect. 3)illustrate where this transition typically occurred.The stabilization phase was defined to begin at the same transition point, and

to end at whichever occurred first of (a) the truck having travelled 250 m afterpassing the lead vehicle, (b) the truck’s longitudinal speed falling below 10 km/h,or (c) the driving simulator’s safety shutdown system having aborted the scenariodue to road departure, or a deviation of truck heading from the road’s forwarddirection of 90◦ or more.

2.4. Model parameter-fitting

The repeated scenario generated 12 measurements for each of the 24 subject driversexcept three, where, due to technical shortcomings, or subject failure to complywith experimental instructions, one or two scenario instances could not be recordedor used. Before parameter-fitting of models, all recorded signals were down-sampledto 5 Hz. The main motivation for down-sampling was the increase in optimizationspeed, but it could also be argued that at high sample rates, adjacent data pointswould anyway be highly correlated, and the input quantities to the tested models,



all constrained by the dynamics of the truck on the road, would also not be expectedto contain much valuable information frequencies above 5 Hz.At each included data point i, a model undergoing parameter-fitting was fed its

required input data from the appropriate recorded signals, with delays if applicable,and the goal of the parameter-fitting, implemented using a genetic algorithm (GA),was to achieve a model output x̂i as close as possible to the observed human controlxi at the same point in time, with xi equal to either δi or δ̇i, depending on themodel1. For further information on GA optimization in general, see [44], and seeAppendix A for full details on the specific GA used here.To quantify model fit, the coefficient of determination R2, interpretable as the

fraction of steering variance being explained by the model [45], was calculated foreach scenario S as:

R2 = 1−SSRSST

= 1−

∑

i∈S(x̂i − xi)2

∑

i∈S(xi − x̄S)2

(6)

where x̄S is the average of xi, for i ∈ S. In other words, R2 can be negative, if a

model provides a worse model fit than simply guessing that x̂i = x̄S for all i ∈ S.Holdout validation [44] was adopted: For each subject driver, the set of available

recorded scenarios was divided into one training set and one validation set, of equalsize2. The GA was set up to maximize the average of R2 across the scenarios in thetraining set, but, in order to prevent over-fitting, the final model parameterizationwas selected as the parameterization with highest average R2 across the validationset. The allocation of recorded scenarios to the two sets was designed to balance theamount of occurring vehicle instability between them: For each driver, the recordedscenarios were ordered by increasing maximum body slip angle, this ordered listwas separated into pairs, and finally one randomly selected scenario in each pairwas assigned to the training set, and the other to the validation set.Thus, for each of the 24 drivers and each of the two steering phases, one opti-

mization was carried out of each tested driver model. The total number of trainingand validation data points xi used for one optimization ranged from 200 (avoid-ance steering for a subject with only ten recorded scenarios) to 900 (stabilizationsteering for a subject with twelve recorded scenarios).

3. Results

Table 1 summarizes the model-fitting results, per model and steering phase, as theaverage validation R2 across the 24 drivers. Also listed are the numbers of effectivefree model parameters (Neff), based on whether parameters were considered to havean effect on steering in the two phases; see Appendix A for full details. Note thatthe open loop avoidance model appears in the table both as a δ and δ̇ controllingmodel (see Sect. 2.2.5).Figs. 6 and 7 show, in their leftmost columns, distributions of per-driver vali-

dation R2 for some of the best-fitting model variants. It can be observed that the

1Alternatively, one could have rerun the studied scenario in closed-loop simulation from initial conditions,fitting parameters to achieve a match between resulting driver steering histories or vehicle trajectories. Suchan approach was not adopted here, both due to it being several orders of magnitude more computation-intensive, and since the inherent instability of the low-friction scenario would presumably have renderedfitting very difficult; a small error in driver model or initial conditions can lead to large deviations inscenario outcome.2Except for one single subject driver where the number of available instances was odd, for which one moreinstance was allocated to the training set.



Table 1. Effective number of model parameters (Neff ) and average goodness-of-fit (R2) on validation data, across

all drivers, for all tested models in the avoidance and stabilization steering phases. Note that some models were

tested in only one of the two phases.

Target signal Model VariantAvoidance Stabilization

Neff Av. R2 Neff Av. R

2

Steering wheel MacAdam 6 0.49 5 0.50angle Sharp et al. 9 0.46 8 0.60

Open loop avoidance constant 3 -1.20looming 4 0.710th order 5 0.541st order 5 0.662nd order 5 0.762nd order, TR fixed 4 0.75

Yaw angle nulling 2 0.35

Steering wheel Salvucci & Gray preview free 8 0.20 8 0.68angle rate preview fixed 6 0.68

Gordon & Magnuski 0th order 5 0.25 4 0.491st order 5 -0.02 4 0.502nd order 5 0.17 4 0.50

Open loop avoidance constant 3 0.41looming 4 0.460th order 5 0.471st order 5 0.402nd order 5 0.47

Yaw rate nulling 2 0.54

spread across drivers was rather similar between models. Further divisions intosubgroups based on driver experience and ESC state indicated limited or no im-pact of these factors on model fit; one example of such a division can be seen inFig. 9(a). Based on these observations, the discussion in the next section will com-pare models mainly in terms of the average validation R2 values in Table 1. As acomplement to this perspective, and to provide a more thorough grasp of actualmodel behaviour, Figs. 6 and 7 also show five example scenario recordings each,along with model predictions. These examples were selected to include driving bothwith and without ESC for both low and high experience drivers, to illustrate somespecific strengths and weaknesses of the different models, while at the same timeaiming for an average R2 across the examples close to the average validation R2

for each model. The discussion in the next section will provide suggestions on howto interpret the various examples.Fig. 8 shows distributions of obtained parameters for the parameter-reduced

variants of the open loop avoidance and Salvucci & Gray models, as well as forthe yaw rate nulling model. The correlation between the safety margin ρC and thesteering gain K in panel (a) is statistically significant1 (r = −0.76; p < 0.0001),but the difference in the steering pulse duration TH between experience groups inpanel (b) is not (t(22) = 0.654; p = 0.51). The correlation between the steeringgains kf and knP in panel (c) is statistically significant (r = −0.71; p = 0.0001), andso is the difference in reaction time TR between experience groups for the Salvucci& Gray model (panel (d); t(22) = −2.19; p = 0.039); however not for the yaw ratenulling model (panel (e); mean TR 0.29 s and 0.34 s for experienced and novicedrivers; t(22) = 1.82; p = 0.083). The correlation between TR and K in panel (e)is statistically significant (r = −0.58; p = 0.003).

1A p < 0.05 criterion for statistical significance is adopted here.

August

10,2014

21:47

Veh

icleSystem

Dynamics

MarkkulaEtA

lVSD

Veh

icleSystem

Dynamics

11

MacAdamN

eff = 6; Av. R2 = 0.49

0 0.5 10

10

20

No. of drivers

−4 −2 0

−100

0

100

R2 = 0.77

Example #1Driver 3 (low exp)

ESC off

−5 −4 −3 −2 −1

−100

0

100

R2 = 0.77


ESC on

−4 −2 0

−100

0

100

R2 = 0.52

Example #3Driver 16 (high exp)

ESC off

−5 −4 −3 −2 −1

−100

0

100

R2 = 0.73


ESC on

−4 −2 0

−100

0

100

R2 = 0.17


ESC off

Human (°)Model (°)

Sharp et al.N

eff = 9; Av. R2 = 0.46

0 0.5 10

10

20

−4 −2 0

−100

0

100

R2 = 0.53−5 −4 −3 −2 −1

−100

0

100

R2 = 0.87−4 −2 0

−100

0

100

R2 = 0.69−5 −4 −3 −2 −1

−100

0

100

R2 = 0.90−4 −2 0

−100

0

100

R2 = −0.12

Open loop avoidance(2nd order, T

R fixed)

Neff

= 4; Av. R2 = 0.75

0 0.5 10

10

20

−4 −2 0

−100

0

100

R2 = −0.23−5 −4 −3 −2 −1

−100

0

100

R2 = 0.90−4 −2 0

−100

0

100

R2 = 0.94−5 −4 −3 −2 −1

−100

0

100

R2 = 0.85−4 −2 0

−100

0

100

R2 = 0.15

Salvucci & Gray(preview free)N

eff = 8; Av. R2 = 0.20

0 0.5 10

10

20

−4 −2 0−200

0

200

R2 = −0.71−5 −4 −3 −2 −1

−200

0

200

R2 = 0.11−4 −2 0

−200

0

200

R2 = 0.26−5 −4 −3 −2 −1

−200

0

200

R2 = 0.36−4 −2 0

−200

0

200

R2 = −0.27

Human (°/s)Model (°/s)

Gordon & Magnuski(0th order)N

eff = 5; Av. R2 = 0.25

0 0.5 10

10

20

−4 −2 0−200

0

200

R2 = −0.28−5 −4 −3 −2 −1

−200

0

200

R2 = 0.41−4 −2 0

−200

0

200

R2 = 0.60−5 −4 −3 −2 −1

−200

0

200

R2 = 0.48−4 −2 0

−200

0

200

R2 = 0.10

Open loop avoidance(2nd order)N

eff = 5; Av. R2 = 0.47

0 0.5 10

10

20

R2−4 −2 0

−200

0

200

R2 = −1.97

Time (s)−5 −4 −3 −2 −1

−200

0

200

R2 = 0.82

Time (s)−4 −2 0

−200

0

200

R2 = 0.70

Time (s)−5 −4 −3 −2 −1

−200

0

200

R2 = 0.43

Time (s)−4 −2 0

−200

0

200

R2 = 0.05

Time (s)

Figure 6. Collision avoidance steering. Distributions of average model performance (leftmost panels) and comparison of human and model steering in five example recorded scenarios,for six of the tested models. Besides including both novices and experienced drivers, driving with and without ESC, the example recordings have been chosen to illustrate typical modelperformance across various steering behaviours, and to highlight strengths and weaknesses of the different models. Note that the top three rows show models controlling steering angle

(δ), whereas the bottom three show models controlling steering rate (δ̇), thus deviating somewhat from the order of model presentation in Sect. 2.2.

August

10,2014

21:47

Veh

icleSystem

Dynamics

MarkkulaEtA

lVSD

12

G.Markku

laet

al.

MacAdamN

eff = 5; Av. R2 = 0.50

0 0.5 10

10

20

No. of drivers

0 2 4 6−1000

0

1000

R2 = 0.65


ESC off

0 2 4 6 8−400−200

0200400

R2 = 0.69


ESC on

0 5 10−50

0

50

R2 = 0.46


ESC off

0 5 10 15−200

0

200

R2 = 0.52


ESC on

0 5 10 15

−200

0

200

R2 = 0.27


ESC off

Human (°)Model (°)

Sharp et al.N

eff = 8; Av. R2 = 0.60

0 0.5 10

10

20

0 2 4 6−1000

0

1000

R2 = 0.830 2 4 6 8

−400−200

0200400

R2 = 0.890 5 10

−50

0

50

R2 = 0.270 5 10 15

−200

0

200

R2 = 0.750 5 10 15

−200

0

200

R2 = 0.31

Yaw angle nullingN

eff = 2; Av. R2 = 0.35

0 0.5 10

10

20

0 2 4 6−1000

0

1000

R2 = 0.450 2 4 6 8

−400−200

0200400

R2 = 0.540 5 10

−50

0

50

R2 = 0.320 5 10 15

−200

0

200

R2 = 0.360 5 10 15

−200

0

200

R2 = 0.17

Salvucci & Gray(preview fixed)N

eff = 6; Av. R2 = 0.68

0 0.5 10

10

20

0 2 4 6

−1000

0

1000

R2 = 0.570 2 4 6 8

−500

0

500

R2 = 0.790 5 10

−200

0

200

R2 = 0.560 5 10 15

−500

0

500

R2 = 0.800 5 10 15

−500

0

500

R2 = 0.52

Human (°/s)Model (°/s)

Gordon & Magnuski(1st order)N

eff = 4; Av. R2 = 0.50

0 0.5 10

10

20

0 2 4 6

−1000

0

1000

R2 = −0.180 2 4 6 8

−500

0

500

R2 = 0.640 5 10

−200

0

200

R2 = 0.550 5 10 15

−500

0

500

R2 = 0.750 5 10 15

−500

0

500

R2 = 0.22

Yaw rate nullingN

eff = 2; Av. R2 = 0.54

0 0.5 10

10

20

R20 2 4 6

−1000

0

1000

R2 = 0.68

Time (s)0 2 4 6 8

−500

0

500

R2 = 0.76

Time (s)0 5 10

−200

0

200

R2 = 0.41

Time (s)0 5 10 15

−500

0

500

R2 = 0.73

Time (s)0 5 10 15

−500

0

500

R2 = 0.27

Time (s)

Figure 7. Stabilization steering. Distributions of average model performance (leftmost panels) and comparison of human and model steering in five example recorded scenarios, forsix of the tested models. Besides including both novices and experienced drivers, driving with and without ESC, the example recordings have been chosen to illustrate typical modelperformance across various steering behaviours, and to highlight strengths and weaknesses of the different models. Note that the top three rows show models controlling steering angle

(δ), whereas the bottom three show models controlling steering rate (δ̇), thus deviating somewhat from the order of model presentation in Sect. 2.2.



0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

ρC (m)

K (

−)

Open loop avoidance (2nd order, TR fixed)

(a)

0 0.2 0.4 0.6 0.8 10

5 Experienced drivers, mean 0.59 s

No.

of d

river

s

0 0.2 0.4 0.6 0.8 10

5 Novice drivers, mean 0.53 s

No.

of d

river

s

TH (s)

(b)

0 5 10 15 200

5

10

15

20

kf (−)

k nI (

1/s)

; knP

(−

)

Salvucci & Gray stabilization (preview fixed)

knP

knI

(c)

0 0.1 0.2 0.3 0.4 0.50

5 Experienced drivers, mean 0.19 s

No.

of d

river

s

0 0.1 0.2 0.3 0.4 0.50

5 Novice drivers, mean 0.24 sN

o. o

f driv

ers

TR (s)

(d)

TR (s)

K (

−)

Yaw rate nulling stabilization

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

10

15

20

25

(e)

Figure 8. Obtained parameter values for the parameter-reduced variants of the open loop avoidance(panels (a) and (b)) and Salvucci & Gray (panels (c) and (d)) models, as well as for the yaw rate nullingmodel (panel (e)), for the 24 drivers in the data collection. In panels (a), (c), and (e), black symbols denoteexperienced drivers, and light red symbols denote novice drivers. Large and small symbols denote driversfor which average model R2 was above and below the median value among the drivers, respectively. Inpanel (c), each driver is represented by a pair of one circle and one square, joined by a vertical line. Thewhite symbols show parameterizations suggested for passenger car driving by Salvucci & Gray [37]; all ofthese are kf = 20, but have been slightly displaced for clarity. In panels (b) and (d), distributions of modelparameters TH (the half-length of the open loop avoidance steering pulse) and TR (the reaction time ofthe Salvucci & Gray model) are shown, separately for experienced and novice drivers.



4. Discussion

In general, the very simple open loop avoidance and yaw rate nulling models turnedout to work rather well, and the performance of the more advanced models canto some extent be understood as being dependent on an ability to generate thebehaviour of the simpler models. These aspects will be discussed below, separatelyfor avoidance and stabilization. Before concluding, some remarks will also be maderegarding the challenges involved in comparing and validating driver behaviourmodels.

4.1. Collision avoidance

4.1.1. Open loop avoidance steering

The open loop avoidance model provided the best fits of the human avoidancesteering, both when comparing among models targeting δ and those targeting δ̇.Although there were cases where the human avoidance steering was more gradual(Example #1 in Fig. 6) or oscillatory (Example #5), in a majority of cases mostof the total steering angle change was applied in a short period of time (Examples#2–#4), such as suggested by the correlation in Fig. 5(a). This type of open-loopaccount of collision avoidance steering is not new, and similar models have beenused not the least in accident reconstruction work and what-if simulations [46–48].The difference in fit between the constant and variable amplitude variants of

the model (especially notable for the δ-controlling model; R2 = −1.20 versus R2

between 0.54 and 0.76) implies that drivers adapted their avoidance steering tothe specific situation. The highest observed R2 values (0.76 and 0.47, for the δand δ̇ controlling variants, respectively) were obtained with the assumption thatdrivers selected their avoidance amplitude based on a 2nd order steering require-ment prediction. This suggests that drivers may have been able to take the non-zerodeceleration of the lead vehicle into account. On the other hand, the model variantsbased on looming, which does not include any acceleration information, reachedalmost as high R2 values (0.71 and 0.46), so the results are far from conclusive inthis respect.When TR was a free parameter in the optimizations, it varied throughout the

entire permitted optimization interval of [0, 2] seconds, something which could beinterpreted as this parameter not being highly important for achieving a good fit,and this is confirmed by the very minor decrease in validation R2 (from 0.76 to0.75) when fixing TR at 0.2 s. It is this parameter-reduced variant of the modelwhich is the basis of Figs. 8(a) and (b). In (a), the correlation between ρC and Kis clearly due to two clusters of parameterizations. These are interpretable, respec-tively, as (1) steering roughly as deemed necessary given the linear, low-frictionvehicle model (K close to 1) to achieve what seems like an unrealistically largesafety margin of about 1.5 – 3 m, and (2) steering aiming for a small safety marginof about 0 – 0.5 m, but applying a steering about three times larger than what thelinear vehicle model predicts would be needed for this purpose. A possible inter-pretation of these fits is that the drivers adapted to the low friction circumstances,responding to vehicle understeering by applying larger steering angles than theywould normally [23, 49]. Indeed, a separate, cursory analysis of the avoidance steer-ing in the first, unexpected scenario suggests that while steering was predominantlypulse-like already at this point in the experiment (as implied also by Fig. 5(a)),the steering pulse amplitudes were generally smaller than predicted by the modelsfitted to the repeated-scenario data. To further clarify exactly how drivers select



their avoidance steering amplitude, more data would be needed, from more variedkinematic situations.With regards to the duration of avoidance steering (parameter TH), Fig. 8(b)

shows considerable variation between individuals, between 0.2 s and 1 s. The steer-ing rate plots in the bottom three rows of Fig. 6 suggest that this is due to variationsin the number of smaller steering corrections needed to achieve satisfactory colli-sion avoidance (cf. [50]). It can also be noted that the average of 0.56 s shown inFig. 8(b) is not far from the 0.42 s predicted by the slope of the correlation inFig. 5. That these values are not exactly identical despite being estimated fromthe same data set is not surprising, if one considers the major differences betweenthe two methods of estimation.

4.1.2. The other models of avoidance steering

In the cases where most of the steering wheel change occurred in a brief periodof time (especially clear in Examples #2 and #3 in Fig. 6), the closed loop δ̇-controlling models by Salvucci & Gray and Gordon & Magnuski were less ablethan the δ̇-controlling open loop model at reproducing the resulting overall pulseof steering change. The closed loop models controlling δ (MacAdam and Sharp etal.) did produce the corresponding step-like δ outputs, but reached lower averagevalidation R2 than the δ-controlling open loop model, despite having a highernumber of free parameters.Besides lower R2 values, another possible objection to the MacAdam and Sharp

et al. models is related to their use of the desired path construct, which in thecontext of collision avoidance could be seen as problematic in at least two ways:(1) With a moving lead vehicle, the desired path will, during a first period of time,typically pass through the lead vehicle, which makes this construct less attractivethan it may seem in scenarios where a path can be charted between stationaryobstacles or lane boundaries. (2) There is a parameter redundancy by which anentire single lane change path (such as in this collision avoidance scenario) canbe shifted longitudinally without affecting the steering behaviour, as long as oneor both of the preview and reaction time parameters are appropriately modifiedat the same time. Besides these specific issues, it can also be noted that recentneurobiological models of basic sensorimotor control seem to be moving away fromdesired trajectory constructs, instead placing emphasis on goal states [51, 52], ar-guably more similar to the target lateral position of the Salvucci & Gray model orthe Gordon & Magnuski model’s goal of avoiding obstacles and lane boundaries.

4.2. Stabilization

4.2.1. Yaw rate nulling stabilization steering

When it comes to stabilization steering, it has been previously shown that modelsfitted to the repeated scenario data could successfully predict also unexpected sce-nario behaviour [32]. Here, the most important new result is the good fits obtainedfor the yaw rate nulling model. Given that the highest average validation R2 acrossall stabilization models was 0.68, for the Salvucci & Gray model with six or eightfree parameters (preview distances fixed or free), the R2 of 0.54 yaw rate nullingmodel, with only two free parameters, seems very good. Qualitatively, the fits (suchas shown in Fig. 7) are also rather convincing. The most natural interpretation ofthese observations seems to be that in the studied scenario, stabilization steeringwas indeed driven to a large extent by a control law similar to what the yaw ratenulling model suggests.Three main types of cases were identified where the yaw rate nulling model did



−0.5 0 0.5 1

10

Yaw rate nulling

Low exp.ESC off

−0.5 0 0.5 1

10

Low exp.ESC on

−0.5 0 0.5 1

10

High exp.ESC off

−0.5 0 0.5 1

10

R2 (−)

High exp.ESC on

Num

ber

of o

bser

vatio

ns (

−)

(a)

101

102

0

0.5

1

R2

(−)

Yaw rate nulling, r = 0.55

101

102

0

0.5

1

R2

(−)

MacAdam, r = −0.04

Maximum yaw rate (°/s)

(b)

2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Scenario repetition (−)

R2

(−)

Yaw rate nulling, r = −0.01

(c)

Figure 9. A more detailed view of model fits for the yaw rate nulling and MacAdam models, as a functionof (a) experimental conditions, (b) maximum yaw rate attained during each scenario, and (c) scenariorepetition. In panels (b) and (c), each small dot corresponds to one recorded scenario from the model-fitting validation set (three points with R2 < 0 not shown for the yaw rate nulling model, two for theMacAdam model), and the r values are the corresponding Pearson correlation coefficients. In (a), the rvalues were calculated with the logarithm of the maximum yaw rates (such as shown here). Without thelogarithms, r = 0.40 (top) and r = 0.05 (bottom). In (b), the rings are averages per repetition.

not work well: (1) Cases where the driver seemingly gave up steering in the faceof imminent control loss (Example #6 in Fig. 7), possibly accounting to some ex-tent for the left tails of the non-ESC distributions in Fig. 9(a), since control losseswere more common without ESC [33]. (2) Cases with less vehicle instability andless critical steering (Example #8 in Fig. 7). Fig. 9(b) illustrates this phenomenonin more detail, by showing increasing model fits for increasing maximum vehicleyaw rates. (3) One or two novice drivers (including driver 6, see Example #10of Fig. 7), who seem to have been using steering strategies of a qualitatively dif-ferent kind, possibly accounting for the low-experience distributions in Fig. 9(a)being marginally farther to the left (combined average R2 = 0.52) than the high-experience distributions (combined average R2 = 0.56).The pattern of lower steering gains K in drivers with longer reaction times TR,

shown in Fig. 8(e), can be interpreted as an adaptation of steering aggressivity toone’s own response speed, to ensure vehicle stability. Such adaptation could haveoccurred as a learning effect during the experiment, but the lack of any clear effectof scenario repetition on model fit (Fig. 9(c)) rather suggests that drivers came tothe experiment with this adaptation already in place.

4.2.2. The other models of stabilization steering

However, the yaw rate nulling model can hardly provide a full account of steeringin the studied scenario; it can stabilize a vehicle directionally, but it has no meansto make it stay on a road or close to some path. In contrast, all of the other testedmodels have such means, and all of them can also be made, more or less naturally,to exhibit some degree of yaw rate nulling.The yaw angle nulling model will, by definition, have a steering rate of the

yaw rate nulling form. Nevertheless, it provides rather poor fits of the humansteering angle data (average validation R2 = 0.35). This could possibly be dueto the model’s lack of a desired path or similar construct, making it unable to



exhibit the rightward lane change during stabilization1. To keep the correspondinglow-frequency component error down (e.g. in the second halves of Examples #8and #9 in Fig. 7), the optimization may have favoured lower steering gains, inturn making the model unable to generate high-frequency steering of sufficientmagnitudes during vehicle instability.If so, the better fits of the Sharp et al. model (average validation R2 = 0.60)

could be due to its yaw angle nulling (the term in eψ) being relative to a desiredpath, but the lateral position error terms in the model may of course also havecontributed. A main source of reduced R2 values for this model seems to have beencases where the driver deviated from his or her own typical path in the scenario,such that the model’s fitted desired path was not appropriate (Example #8).The MacAdam model also has a desired path, but no direct means of applying

yaw angle or yaw rate nulling. The fits obtained here had long preview times TP(average 3.6 s, compared to 2.6 s for the Sharp et al.model). This makes the optimalcontrol prioritize following the general direction of the road over correcting forlocal lateral position errors, in essence reducing it to the yaw angle nulling model,a similarity which is clear from Fig. 7. The resulting validation R2 average of 0.50was slightly lower than for the yaw rate nulling model, despite the larger numberof free parameters, and there was no increase in model fit with increasing yawinstability (Fig. 9(b)).The Gordon & Magnuski model, by Eq. (5), applies yaw rate nulling as long as

|ψ̇req| ≪ |ψ̇|. However, when close to a lane exceedence (Example #6 in Fig. 7)the model prioritizes lane keeping higher than the human drivers did. Overall, themodel is also less aggressive in its yaw rate nulling behaviour than the humans (seethe other examples in Fig. 7); possible reasons for this include the steering gainsbeing kept down (the τs being kept high) to minimize error when yaw rates are low,and the model’s satisficing approach of aiming for non-zero yaw rate remainderswhere the humans seemingly did not.As shown mathematically in [32] and illustrated in Fig. 10, on a straight road

far point rotation approximates negative yaw rate, such that the far point controlof the Salvucci & Gray model can be understood as yaw rate nulling1. This insighthelps explain the success of the Salvucci & Gray model in fitting the stabilizationsteering data, and also provides a candidate for a perceptual cue supporting yawrate nulling behavior. However, the fact that the far point was parameter-fitted,here, to Df = 123 m ahead of the truck, whereas the 3

◦ down from the horizonsuggested by previous authors [37, 40] correspond to Df ≈ 50 m for the truck inthe experiment, could be taken to suggest that also other cues, such as vestibularcues [23] or large-field visual motion [55] may have been at play.With regards to the other parameters of the Salvucci & Gray model, it is in-

teresting to note the statistically significant faster response times for experienceddrivers (Fig. 8(d)), in line with what has been suggested by several other authors;see e.g. [23] and [20, pp. 1132–1133]. The correlation between kf and knP (Fig. 8(c))could be understood as a parameter redundancy; one which is not surprising giventhe strong correlation between near and far point rates visible in Fig. 10. It is clearthat with Dn = 16 m, also the near point angle rate was a very close approximationof negative yaw rate, especially at low lateral speeds relative to the road. The exactvalues obtained here for kf and knP should therefore not be attributed too muchimportance; they could simply be a more or less arbitrary division of the yaw ratenulling model’s single gain parameter K.

1For studies of a yaw angle nulling model with a desired path, see [53] or [54].1On a circular road, far point rotation nulling corresponds to nulling of yaw rate error.



0 50 100 150 200 250

−10

−8

−6

−4

−2

Late

ral p

ositi

on (

m)

0 50 100 150 200 250

−20

0

20


Near point angle (°)Near point angle rate (°/s)Far point angle rate (°/s)Negative yaw rate (°/s)

Figure 10. An illustration of the input quantities used by the Salvucci & Gray [37] model, in one examplerecording. The top panel shows the vehicle trajectory, including arrows showing momentary heading of thetruck’s front. The bottom panel shows the model input quantities, for Dn = 16 m and Df = 123 m, aswell as negative vehicle yaw rate. The discontinuity in the near point angle plot corresponds to the truckreaching longitudinal position X3, where the model switches its target lane to the right. In both panels,longitudinal position zero corresponds to the point where the truck’s front reached the rear of the leadvehicle, and the vertical dashed lines show the beginning and end of the stabilization phase as defined inSec. 2.3.

4.3. Comparing models of driver control behaviour

One limitation in the comparisons presented here arises from some of the modelspredicting steering wheel angle δ, and others its time derivative δ̇. The differenti-ation from δ to δ̇ attenuates steering variations at low frequencies and amplifiesthose at high frequencies, which means that model-fitting to these two signals willput emphasis on different aspects of steering. For example, the δ and δ̇-controllingvariants of the open loop avoidance model are logically equivalent, but the lim-itations of assuming a single burst of steering are more obvious in the δ̇ signalthan in the δ signal, and this results in lower R2 values for the δ̇ model variants.Indeed, none of the tested δ̇-controlling models include any input signals or mecha-nisms which could have allowed them to fully reproduce the type of high-frequencyvariations in δ̇ visible in Figs. 6 and 7.Another limitation, here, is the informal treatment of model parameter count.

In theory, additional model parameters cannot reduce model fit, only increase it,and will at the same time increase the risk of obtaining parameter values whichoverfit to regularities that are unique to the specific data set at hand (e.g. due toonly considering one single driving scenario, such as here), thus potentially reducinggenerality of the parameter-fitted model. There are statistical methods for properlymanaging this trade-off between model complexity and model fit [56], but theseare devised for probabilistic models, as opposed to the completely deterministicmodels considered here.Because of the limitations outlined above, the results presented here do not

provide grounds for a conclusive recommendation on what model or models toprefer for e.g. simulated evaluation of ESC. Leaving between-model R2 comparisonsto the side, an advantage of the Sharp et al. model is that it performed reasonablywell both in avoidance and stabilization, implying that one could use a single modelfor an entire scenario. On the other hand, it could be argued that the Salvucciand Gray model seems more psychologically plausible, due to its input quantitiesbeing readily available to a human driver, and since it does not need to assumean internally planned desired path. Psychological plausibility may not be a majorpriority in some applied contexts, but could provide the benefit of a model thatgeneralizes better beyond the specific data to which it has been parameter-fitted.It should also be acknowledged, however, that while parameter-fitting of models



Table 2. A summary of advantages and disadvantages of the various tested models, in the studied low-friction

collision avoidance scenario.

Model Advantages in the studied scenario Disadvantages in the studied sce-nario

MacAdam Vehicle-independent parameters.Reasonable fit of avoidance steering.

Relies on desired path construct toachieve fit of avoidance steering.Rather poor fit of stabilization steer-ing.

Sharp et al. Reasonable fits of both avoidanceand stabilization steering.

Relies on desired path construct.Many parameters.

Salvucci & Gray Best fit of stabilization steering.Psychologically plausible inputquantities.

Poor fit of avoidance steering.Rather many parameters.

Gordon & Magnuski Psychological plausibility of satisfic-ing approach.

Poor fit of avoidance steering andrather poor fit of stabilization steer-ing.

Open loop avoidance Best fit of avoidance steering. Not applicable to stabilization steer-ing.

Yaw angle nulling Few parameters. Not applicable to avoidance steering.Poor fit of stabilization steering. Notuseful as a closed-loop model.

Yaw rate nulling Reasonable fit of stabilization steer-ing with few parameters, thus po-tentially indicative of a relevant be-havioural phenomenon.

Not applicable to avoidance steering.Not useful as a closed-loop model.

such as performed here may be useful for understanding differences between models,and for pruning out models which do not work at all, it is not necessarily a suitablemethod for elucidating psychological mechanisms [57]. Here, the good fits of theyaw rate nulling model seem rather compelling, since this model has so few freeparameters, but due to the parameter count effects discussed above, the higher R2

values for the six-parameter Salvucci & Gray model should not be taken as proofof that model’s underlying assumptions, e.g. that drivers are using near and farpoints to guide their steering. To study underlying mechanisms, a better approachis to instead identify situations where competing models diverge in their predictionsabout human behaviour, and then test these predictions in experiment [57].In future model comparisons, to avoid the δ-δ̇ type of difficulty, one could con-

sider fitting all models to the same control signal (e.g. δ̇). A study of closed-loopbehaviour of the parameter-fitted models is also a natural next step, but has beenbeyond the scope here.

5. Conclusion

The work presented here has clarified some similarities and differences between anumber of existing and novel models of driver steering. The strengths and weak-nesses of these models in the specific studied scenario are summarized in Table 2.While it has been shown that several of the tested models were reasonably capa-ble of reproducing the observed human steering behaviour, it also remains clearthat, even within a well-defined and constrained context, it is non-trivial to decidewhich exact models to prefer over others. Furthermore, the poor fits reported herefor some models do not imply that these models cannot work well in other contextsor scenarios. Especially regarding the Salvucci & Gray and Gordon & Magnuskimodels, it should be acknowledged that they were originally formulated for routinelane keeping, rather than near-limit manoeuvring.Overall, model fits were not much affected by whether drivers were novices or

experienced, or whether they were driving with ESC on or off. The drivers includedhere were all truck drivers, driving a simulated truck, but the simplicity of themain observed behavioural phenomena (open loop avoidance and yaw rate nulling



stabilization) makes it reasonable to hypothesize that these phenomena could occuralso in passenger car driving.Regarding the general approach to modelling human control behaviour, the vari-

ous models tested here are based on rather different underlying assumptions: Fromthe MacAdam model that emphasizes an internalized model of vehicle dynamicsand a desired path, to the Salvucci & Gray model that instead emphasizes vi-sual cues allowing the driver to aim for a target lateral position. The results andanalyses presented here show that these types of accounts can predict equivalentbehaviour in some cases, but could diverge in others:Collision avoidance steering was best described by the open-loop model, and

the best-fitting version (average validation R2 = 0.76) could be interpreted asthe drivers acquiring an updated internal model of the vehicle’s behaviour on thelow-friction road. However, the version of the same model based instead on opti-cal expansion of the lead vehicle on the driver’s retina performed almost as well(average validation R2 = 0.71), with one free parameter less.Stabilization steering was explained to a large extent (average validation R2 =

0.54), and especially well in recordings with more pronounced yaw instability, bythe two-parameter yaw rate nulling model, according to which drivers apply asteering rate proportional to the negative of the vehicle’s yaw rate. It is interestingto note that the Salvucci & Gray model (and to some extent also the Sharp et al.and Gordon & Magnuski models) clearly does predict a causation from instabilityto yaw rate nulling, due to sight point rotation nulling, and the Salvucci & Graymodel also provided the highest average validation R2 of 0.68 for the stabilizationsteering data.The MacAdam model on the other hand, despite having more free parameters

than the yaw rate nulling model, provided slightly worse fits of the stabilization data(average validation R2 = 0.50, without a trend of better fits for more pronouncedyaw instability). While, again, it is possible that better fits could be obtained byassuming that drivers acquired a more advanced, non-linear internal vehicle model[23, 49], compensating for tyre saturation with increased steering, it is presentlynot clear whether additional layers of assumed driver insight into vehicle dynamicswould in the end really result in such a simply described, and seemingly non-optimal, behaviour as yaw rate nulling.The fact that yaw rate nulling behavior during instability is correctly predicted

by models originally devised for non-critical driving suggests the interesting possi-bility that drivers may be applying the same sensorimotor control heuristics (e.g.sight point rotation nulling) in both routine and critical situations (cf. [58]). By suchan account, seemingly optimal, vehicle-dynamics-adapted behaviour from driversin routine driving situations can be understood as these sensorimotor heuristics,although far from optimal in general, being precisely tuned for performance andefficiency after extended practice in a constrained operating regime. This would im-ply that modellers can afford themselves the practical advantages of optimal controltheory and internal vehicle dynamics representations, when simulating driving sit-uations of which the modelled driver has much experience (e.g. normal driving fornormal drivers, race car driving for race car drivers). However, when modelling lessfrequent situations, such as traffic near-crashes, one may be better off with a modelthat is based on the underlying sensorimotor heuristics.

Acknowledgments

This work was supported by VINNOVA (the Swedish Governmental Agency forInnovation Systems; grant number 2009-02766). The data collection was also sup-


REFERENCES 21

ported by the competence centre ViP (Virtual Prototyping and Assessment bySimulation, www.vipsimulation.se, co-financed by the VINNOVA grant 2007-03083 and the competence centre partners).

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Table A1. A listing of all model parameters included in the model-fitting optimizations. For each parameter,

the allowed range in the optimizations is indicated, together with information on what models made use of

the parameter, and on whether the model was considered effective in the collision avoidance and stabilization

steering phases, respectively (i.e. whether or not it was included in the corresponding Neff count in Table 1).

Parameter Allowed range Used by modelsa Collision avoidance Stabilization

∆X1 [−50, 50] m M, S xX2 [−20, 30] m M, S xX3 [20, 70] m M, S, S&G x xX4 [100, 250] m M, S, G&M xY [−1.65,−6.5] m M, S, S&G x xTP [0.1, 5] s M, S x xTR [0, 2] s M, S, S&G, G&M, OLA (ex-

cept constant amplitude vari-ant), YAN, YRN

x x

Kψ [0, 100] S x xK1 [0, 100] S x xKp [0, 10] S x x∆XI [−50, 0] m S&G, G&M xDn [0.1, 200] m S&G x xDf [0.1, 200] m S&G x xkf [0, 100] S&G x xknP [0, 50] S&G x xknI [0, 10] S&G x xρC [0, 3] m G&M, OLA (steering require-

ment variants)x x

ρL [−2.25, 3] m G&M x xτs [0, 10] s G&M x xK [0, 20] OLA (constant amplitude vari-

ant)x N/A

K [0, 200] OLA (other variants) x N/ATA [−0.5, 0.5] s OLA x N/ATH [0.1, 1] s OLA x N/AK [0, 100] YAN, YRN N/A x

aM: MacAdam; S; Sharp et al.; S&G: Salvucci & Gray; G&M: Gordon & Magnuski; OLA: Open loopavoidance; YAN: Yaw angle nulling; YRN: Yaw rate nulling.

Appendix A. Genetic algorithm implementation

In the GA used for model-fitting (see Sec. 2.4), a candidate model parameterizationwas represented by a GA individual with a genome of length N , the number offree parameters of the model. Each gene was a floating point number in the inter-val [0, 1], corresponding to a value within the allowed range for the parameter inquestion. These ranges are listed in Table A1, also showing which parameters wereconsidered effective in the collision avoidance and stabilization phases, respectively.The GA was configured, in the terminology and notation of [44, pp. 48–55], as

follows: population size 100, tournament size 2, tournament selection parameterptour = 0.9, crossover probability pc = 1, mutation probability pmut = 1/N . Mu-tation consisted in either (with probability 0.5) randomly choosing a new valuefrom a uniform distribution in [0, 1], or otherwise applying real-number creep froma normal distribution of standard deviation 0.005, after which the new value wasbounded to [0, 1]. The best individual in a given GA generation was always retainedin the next generation (elitism). Initial exploration indicated that model-fitting R2

values were not very sensitive to the exact GA configuration, and the specific GAparameter settings adopted here were selected based on a criterion of low variabilityin R2 estimates across repeated optimizations.The GA was terminated at completion of generation number 2G, whereG was the

last generation in the optimization with an increase in validation fitness. However,all optimizations were allowed a minimum of 300 generations.

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